BOLTZMANN CONSTANT: HISTORY, EQUATIONS, CALCULUS, EXERCISES - PHYSICAL - 2025

The Boltzmann constant is the value that relates the average kinetic energy of a thermodynamic system or an object with the absolute temperature of the same. Although they are often confused, temperature and energy are not the same concept.

Temperature is a measure of energy, but not energy itself. With Boltzmann's constant they are linked to each other in the following way:

Boltzmann's tombstone in Vienna. Source: Daderot at English Wikipedia

This equation is valid for a monatomic ideal gas molecule of mass m, where E _c is its kinetic energy given in Joules, k _B is Boltzmann's constant and T is the absolute temperature in Kelvin.

In this way, when the temperature increases, the average kinetic energy per molecule of substance also increases, as is expected to happen. And the opposite happens when the temperature decreases, being able to reach the point where if all movement stops, the lowest possible temperature or absolute zero is reached.

When talking about average kinetic energy, it is necessary to remember that kinetic energy is associated with motion. And particles can move in many ways, such as moving, rotating, or vibrating. Of course, they will not all do it in the same way, and since they are uncountable, then the average is taken to characterize the system.

Some energy states are more likely than others. This concept is of radical importance in thermodynamics. The energy considered in the previous equation is translational kinetic energy. The probability of states and its relationship with Boltzmann's constant will be discussed a little later.

In 2018 the Kelvin was redefined and with it the Boltzmann constant, which in the International System is approximately 1.380649 x 10 ^-23 J. K ^-1. Much more precision can be achieved for the Boltzmann constant, which has been determined in numerous laboratories around the world, by different methods.

History

The famous constant owes its name to the Vienna-born physicist Ludwig Boltzmann (1844–1906), who dedicated his life as a scientist to the study of the statistical behavior of systems with many particles, from the point of view of Newtonian mechanics.

Although today the existence of the atom is universally accepted, in the 19th century the belief about whether the atom really existed or was an artifice with which many physical phenomena were explained was in full debate.

Boltzmann was a staunch defender of the existence of the atom, and in his time faced harsh criticism of his work from many colleagues, who considered it to contain insoluble paradoxes.

He stated that observable phenomena at macroscopic levels could be explained by the statistical properties of constituent particles like atoms and molecules.

It may be that these criticisms were due to the profound episode of depression that led him to take his life at the beginning of September 1906, when he still had a lot to do, since he was considered one of the great theoretical physicists of his time and there was very little left to go. that other scientists contribute to corroborate the veracity of their theories.

It was not long after his death that new discoveries about the nature of the atom and its constituent particles added to prove Boltzmann right.

Boltzmann's constant and Planck's works

Now the Boltzmann constant k _B was introduced as it is known today some time after the work of the Austrian physicist. It was Max Planck, in his law of the emission of the black body, a work that he presented in 1901, who at that time gave it the value of 1.34 x 10 ⁻²³ J / K.

Around 1933 a plaque with the definition of entropy involving the famous constant was added to Boltzmann's tombstone in Vienna as a posthumous tribute, which involves the famous constant: S = k _B log W, an equation that will be discussed later.

Today Boltzmann's constant is indispensable in the application of the laws of thermodynamics, statistical mechanics and information theory, fields of which this sad-ending physicist was a pioneer.

Value and equations

Gases can be described in macroscopic terms and also in microscopic terms. For the first description there are concepts such as density, temperature and pressure.

However, it should be remembered that a gas is composed of many particles, which have a global tendency to a certain behavior. It is this trend that is measured macroscopically. One way to determine the Boltzmann constant is thanks to the well-known ideal gas equation:

Here p is the pressure of the gas, V is its volume, n is the number of moles present, R is the gas constant and T is the temperature. In a mole of ideal gas, the following relationship is fulfilled between the product pV, and the translational kinetic energy K of the whole set is:

Therefore the kinetic energy is:

By dividing by the total number of molecules present, which will be called N, the average kinetic energy of a single particle is obtained:

In one mole there is Avogadro's number of particles N _A, and therefore the total number of particles is N = nN A, leaving:

Precisely the quotient R / N _A is the Boltzmann constant, thus being shown that the average translational kinetic energy of a particle depends only on the absolute temperature T and not on other quantities such as pressure, volume or even the type of molecule:

Boltzmann's constant and entropy

A gas has a given temperature, but that temperature can correspond to different states of internal energy. How to visualize this difference?

Consider the simultaneous flip of 4 coins and the ways in which they can fall:

Ways in which 4 can drop 4 coins. Source: self made

The set of coins can assume a total of 5 states, which are considered macroscopic, described in the figure. Which of these states would the reader say is the most likely?

The answer should be the state of 2 heads and 2 tails, because you have a total of 6 possibilities, of the 16 illustrated in the figure. Y 2 ⁴ = 16. These equal the microscopic states.

What if 20 coins are tossed instead of 4? There would be a total of 2 ²⁰ possibilities or "microscopic states". It is a much larger number and more difficult to handle. To facilitate the handling of large numbers, logarithms are very appropriate.

Now, what does seem evident is that the state with the greatest disorder is the most likely. More ordered states like 4 heads or 4 seals are slightly less likely.

The entropy of a macroscopic state S is defined as:

Where w is the number of possible microscopic states of the system and k _B is Boltzmann's constant. Since ln w is dimensionless, the entropy has the same units as k _B: Joule / K.

This is the famous equation on Boltzmann's tombstone in Vienna. However, more than entropy, what is relevant is its change:

How do you calculate k

The value of Boltzmann's constant is obtained experimentally in an extremely precise way with measurements based on acoustic thermometry, which are carried out using the property that establishes the dependence of the speed of sound in a gas with its temperature.

Indeed, the speed of sound in a gas is given by:

B _adiabatic = γp

And ρ is the density of the gas. For the above equation, p is the pressure of the gas in question and γ is the adiabatic coefficient, whose value for a given gas is found in tables.

Metrology institutes are also experimenting with other ways to measure the constant, such as the Johnson Noise Thermometry, which uses random thermal fluctuations in materials, particularly conductors.

Solved exercises

-Exercise 1

Find:

a) The average translational kinetic energy E _c that an ideal gas molecule has at 25 ºC

b) The translational kinetic energy K of the molecules in 1 mole of this gas

c) The average speed of an oxygen molecule at 25 ºC

Fact

m _oxygen = 16 x 10 ^-3 kg / mol

Solution

a) E _c = (3/2) k T = 1.5 x 1.380649 x 10 ^-23 J. K ^-1 x 298 K = 6.2 x 10 ^-21 J

b) K = (3/2) nRT = 5 x 1 mol x 8.314 J / mol. K x 298 K = 3716 J

c) E _c = ½ mv ², taking into account that the oxygen molecule is diatomic and the molar mass must be multiplied by 2, we will have:

Find the change in entropy when 1 mole of gas occupying a volume of 0.5 m ³ expands to occupy 1 m ³.

Solution

ΔS = k _B ln (w ₂ / w ₁)

References

1. Atkins, P. 1999. Physical Chemistry. Omega editions. 13-47.
2. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill. 664- 672.
3. Giancoli, D. 2006. Physics: Principles with Applications. 6th.. Ed Prentice Hall. 443-444.
4. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 1. 647-673.
5. YES Redefinition. Kelvin: Boltzmann Constant. Retrieved from: nist.gov